L(s) = 1 | + (−0.342 − 0.939i)5-s + (0.342 − 0.939i)11-s + (0.342 + 0.939i)13-s + (−0.5 − 0.866i)17-s + (0.866 + 0.5i)19-s + (−0.173 + 0.984i)23-s + (−0.766 + 0.642i)25-s + (−0.342 + 0.939i)29-s + (−0.939 + 0.342i)31-s + i·37-s + (0.939 − 0.342i)41-s + (−0.984 + 0.173i)43-s + (−0.939 − 0.342i)47-s + (−0.866 − 0.5i)53-s − 55-s + ⋯ |
L(s) = 1 | + (−0.342 − 0.939i)5-s + (0.342 − 0.939i)11-s + (0.342 + 0.939i)13-s + (−0.5 − 0.866i)17-s + (0.866 + 0.5i)19-s + (−0.173 + 0.984i)23-s + (−0.766 + 0.642i)25-s + (−0.342 + 0.939i)29-s + (−0.939 + 0.342i)31-s + i·37-s + (0.939 − 0.342i)41-s + (−0.984 + 0.173i)43-s + (−0.939 − 0.342i)47-s + (−0.866 − 0.5i)53-s − 55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.135 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.135 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4370817042 + 0.5010167310i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4370817042 + 0.5010167310i\) |
\(L(1)\) |
\(\approx\) |
\(0.8705094914 - 0.05354035366i\) |
\(L(1)\) |
\(\approx\) |
\(0.8705094914 - 0.05354035366i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-0.342 - 0.939i)T \) |
| 11 | \( 1 + (0.342 - 0.939i)T \) |
| 13 | \( 1 + (0.342 + 0.939i)T \) |
| 17 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 + (0.866 + 0.5i)T \) |
| 23 | \( 1 + (-0.173 + 0.984i)T \) |
| 29 | \( 1 + (-0.342 + 0.939i)T \) |
| 31 | \( 1 + (-0.939 + 0.342i)T \) |
| 37 | \( 1 + iT \) |
| 41 | \( 1 + (0.939 - 0.342i)T \) |
| 43 | \( 1 + (-0.984 + 0.173i)T \) |
| 47 | \( 1 + (-0.939 - 0.342i)T \) |
| 53 | \( 1 + (-0.866 - 0.5i)T \) |
| 59 | \( 1 + (-0.642 + 0.766i)T \) |
| 61 | \( 1 + (-0.342 + 0.939i)T \) |
| 67 | \( 1 + (-0.984 - 0.173i)T \) |
| 71 | \( 1 + (0.5 - 0.866i)T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + (0.173 + 0.984i)T \) |
| 83 | \( 1 + (-0.342 + 0.939i)T \) |
| 89 | \( 1 + (0.5 - 0.866i)T \) |
| 97 | \( 1 + (0.173 + 0.984i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−18.830459711413145937512534731343, −18.00504391686418907884572153711, −17.74187059007844074051477004443, −16.84118889898450604165812192396, −15.84320419149915116175549336641, −15.34971110252340045260299719540, −14.70048596568924561149731143582, −14.15594227023462914009628713256, −13.08158818961825170070849095437, −12.63513742821327240715151829161, −11.65568572268418557134207496182, −11.05573668872703816511747826114, −10.38402149485231457443995924328, −9.70315665522969624080107849333, −8.85735077460524579299298569898, −7.82180649359884314166145339547, −7.465947834738901652954771478787, −6.47172416939206284883708075611, −5.98603096896826063617565241345, −4.85679325543530484380736618773, −4.05913421954691771610538289123, −3.317726430346048610299701928259, −2.470384824152935661015385733097, −1.63583866134453387858642413886, −0.20623815085944364522487542463,
1.16527893243295345679883413456, 1.71182777588444753750222315108, 3.14541541307685371884931707714, 3.71279085622859157007365664713, 4.64478257160265364736435512285, 5.325487276958099506724491910842, 6.09675674470797909799623071733, 7.04700931786736154660864544364, 7.759055398809841600517460026399, 8.65586620937252552949141384474, 9.13887036128448833240955059896, 9.75373005093434361718936463922, 10.96524899021600073474872659486, 11.574745884293956335147164866479, 11.99935039318712976360519793676, 12.99240133316228177904797014629, 13.64967257954333056773285064033, 14.14642612435292776990443480370, 15.12546189175348168763961602428, 16.07216911772015195088856804852, 16.33274856370942695225919525486, 16.90730001401336279643281705227, 17.96999136636649843675736856164, 18.476562393591058756295292277357, 19.40116295617724272238610021827